This application claims the priority of Japanese Patent Application No. 2000-092714 filed on Mar. 30, 2000 and Japanese Patent Application No. 2000-359142 filed on Nov. 27, 2000, which are incorporated herein by reference.
1. Field of the Invention
The present invention relates to a fringe analysis method and apparatus using Fourier transform; and, in particular, to a fringe analysis method and apparatus which can effectively use Fourier transform method when analyzing image data having fringe patterns such as interference fringe patterns.
2. Description of the Prior Art
Light-wave interferometry has conventionally been known as important means concerning accurate measurement of object wavefront. In recent years, there has been an urgent need for developing an interferometry technique (sub-fringe interferometry) for reading out information from a fraction of a single interference fringe (one fringe) or less from the necessity to measure a surface or wavefront aberration at an accuracy of {fraction (1/10)} wavelength or higher.
For sub-fringe interferometry techniques, attention has been focused on techniques using Fourier transform method as disclosed in xe2x80x9cBasics of Sub-fringe Interferometry,xe2x80x9d Kogaku, Vol. 13, No. 1 (February, 1984), pp. 55-65, for example.
However, the fringe analysis method (Fourier transform fringe analysis method) using Fourier transform, which is excellent in principle, leaves some problems unsolved and has not always been effectively put into practice.
One of these problems lies in that a large analysis error may occur in the Fourier transform method when introducing a carrier frequency.
The Fourier transform fringe analysis method will now be explained.
Fourier transform fringe analysis method is a technique which makes it possible to determine the phase of a wavefront with high accuracy from a single sheet of fringe image by introducing a carrier frequency (caused by an inclination of a surface to be measured or a reference). When the carrier frequency is introduced, the interference fringe intensity is represented by the following expression (1):
i(x, y)=a(x, y)+b(x, y)cos(2xcfx80fxx+2xcfx80fyy+xcfx86(x, y))xe2x80x83xe2x80x83(1)
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
xcfx86(x, y) is the phase of the wavefront; and
fx and fy are the respective carrier frequencies in the x and y directions represented by:             f      x        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          x                    λ        ,            f      y        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          y                    λ      
where xcex is the wavelength of light, and xcex8x and xcex8y are the respective inclinations of the surface to be observed in the x and y directions.
The above-mentioned expression (1) can be converted into the following expression (2):
i(x, y)=a(x, y)+c(x, y)exp[i(2xcfx80fx+2xcfx80fy)]+c*(x, y)exp[i(2xcfx80fx+2xcfx80fy)]xe2x80x83xe2x80x83(2)
where c(x, y) is the complex amplitude of interference fringes, and c*(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented as the following expression (3):                               c          ⁡                      (                          x              ,              y                        )                          =                                            b              ⁡                              (                                  x                  ,                  y                                )                                      ⁢                          exp              ⁢                              xe2x80x83                            [                              ⅈφ                ⁡                                  (                                      x                    ,                    y                                    )                                            ]                                2                                    (        3        )            
The Fourier transform of expression (2) gives:
I(xcex7,xcex6)=A(xcex7,xcex6)+C(xcex7xe2x88x92fx,xcex6xe2x88x92fy)+C*(xcex7xe2x88x92fx,xcex6xe2x88x92fy)xe2x80x83xe2x80x83(4)
where A(xcex7,xcex6) is the Fourier transform of a(x, y), whereas C(xcex7xe2x88x92fx, xcex6xe2x88x92fy) and C*(xcex7xe2x88x92fx, xcex6xe2x88x92fy) and C*(xcex7xe2x88x92fx, xcex6xe2x88x92fy) are the Fourier transforms of c(x, y) and c*(x, y), respectively.
Subsequently, C(xcex7xe2x88x92fx, xcex6xe2x88x92fy) is taken out by filtering, and the peak of the spectrum located at coordinates (fx, fy) is transferred to the origin of a frequency coordinate system (also referred to as Fourier spectra plane coordinate system), so as to eliminate the carrier frequencies. Then, c(x, y) is determined by use of inverse Fourier transform, and the wrapped measured phase can be obtained by:                               φ          ⁡                      (                          x              ,              y                        )                          =                              Im            ⁡                          (                              c                ⁡                                  (                                      x                    ,                    y                                    )                                            )                                            Re            ⁡                          (                              c                ⁡                                  (                                      x                    ,                    y                                    )                                            )                                                          (        5        )            
where Im(c(x,y)) is the imaginary part of c(x,y) and Re(c(x,y)) is the real part of c(x,y).
Finally, unwrapping processing is carried out, so as to determine the phase xcfx86(x, y) of the wavefront to be measured.
In the Fourier transform fringe analysis method explained in the foregoing, while the fringe image data modulated by a carrier frequency is subjected to Fourier transform as mentioned above, a large error is often included in the result of arithmetic operation at this time in practice. The error in analysis may extend to about a few percent of the wavelength, thereby becoming a major factor which hinders such a technique from being put into practice.
Therefore, it is a first object of the present invention to provide a fringe analysis method and apparatus using Fourier transform method which can efficiently determine the posture of the object according to fringe image data in which a carrier frequency is introduced.
It is a second object of the present invention to provide a fringe analysis method and apparatus which can minimize errors in arithmetic operations when fringe image data in which a carrier frequency is introduced is subjected to Fourier transform method, thereby yielding favorable analysis results with less errors.
The present invention provides a fringe analysis method using Fourier transform, in which fringe image data carrying wavefront information of an object is obtained according to a relative wavefront profile of the object with respect to a reference, the fringe image data being in a state where a wavefront from the object and a wavefront from the reference are relatively inclined by a minute amount with respect to each other and a carrier fringe occurring due to the inclination is superposed on a fringe occurring due to the wavefront information of the object;
the method comprises the step of subjecting the fringe image data to Fourier transform method so as to determine a wavefront profile of the object;
the inclination is set such that a carrier frequency occurring due to the inclination is a predetermined multiple of the basic frequency determined by the wavefront information of the object and observing means.
Preferably, in the fringe analysis method using Fourier transform in accordance with the present invention, the inclination is set such that a carrier frequency occurring due to the inclination is substantially an integral multiple of the basic frequency determined by the wavefront information of the object and observing means.
The present invention provides a fringe analysis apparatus using Fourier transform method, in which fringe image data carrying wavefront information of an object obtained according to a relative wavefront profile of the object with respect to a reference is subjected to Fourier transform method so as to determine a wavefront profile of the object;
the apparatus comprises:
an inclination adjusting mechanism for adjusting a relative inclination of a wavefront from the object and a wavefront from the reference with respect to each other; and
an inclination adjusting mechanism driving means for driving the inclination adjusting mechanism such that a carrier frequency occurring due to the inclination is a predetermined multiple of the basic frequency determined by the wavefront information of the object and observing means.
Preferably, in the fringe analysis apparatus using Fourier transform method in accordance with the present invention, the inclination adjusting mechanism driving means drives the inclination adjusting mechanism such that a carrier frequency occurring due to the inclination is substantially an integral multiple of the basic frequency determined by the wavefront information of the object and observing means.
In an embodiment in this case, the inclination adjusting mechanism adjusts a relative inclination of the object and the reference with respect to each other, and the inclination adjusting mechanism driving means drives the inclination adjusting mechanism such that a carrier frequency occurring due to the inclination is substantially an integral multiple of the basic frequency determined by the wavefront information of the object and observing means.
Though not restrictive, the inclination adjusting mechanism may comprise members composed of one fulcrum member and two piezoelectric actuators or three piezoelectric actuators for inclining the object or a reference member provided with the reference, the members being arranged such that two lines on the reference member connecting the fulcrum member to the respective piezoelectric actuators are orthogonal to each other; or may have a tube-shaped piezoelectric actuator, adapted to incline in two axial directions, for driving the object or a reference member provided with the reference.
The fringe analysis apparatus using Fourier transform method may comprise:
imaging means for capturing the fringe image;
carrier frequency calculating means for calculating according to the captured fringe image data a carrier frequency occurring in response to a relative inclination of the object and the reference with respect to each other;
frequency difference calculating means for calculating a difference between the carrier frequency becoming an integral multiple of the basic frequency determined by the wavefront information of the object and observing means, and the calculated carrier frequency; and
inclination amount adjusting means for calculating a relative inclination amount of the object and the reference with respect to each other which is required for correcting the difference between frequencies, and sending out a signal corresponding to thus calculated inclination amount to the inclination adjusting mechanism driving means;
so as to feedback-control the relative inclination amount of the object and the reference with respect to each other.
In the calculation carried out by the carrier frequency calculating means, the captured fringe image data may be subjected to Fourier transform method, so as to determine a carrier frequency value (fx, fy) according to positional coordinates of a maximum peak excluding a peak located at the origin among peaks on a frequency coordinate system obtained by the Fourier transform method.
Here, the basic frequency is represented by the following expressions:
basic frequency in x direction fsx=1/Nx
basic frequency in y direction fsy=1/Ny
where Nx is the number of sampling points in x direction, and Ny is the number of sampling points in y direction.
The fringe image data may be interference fringe image data.
The fringe image data may be captured by use of a Michelson type interferometer.
Here, the wavefront information of the object does not include the wavefront profile caused by the relative inclination of the object and reference with respect to each other that is provided for generating the carrier frequency.
Furthermore, the wavefront information may be surface profile information of the object.